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Hii. Let me answer your question!!!

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Let's solve this problem step-by-step:

1. Calculate the volume of each cone.

The volume of a cone is given by:

[tex]V_{cone} = \frac{1}{3} \pi r^{2} h\: units^{3}[/tex]

where:

• [tex]V_{cone}[/tex] is the volume of the cone
• [tex]\pi[/tex] is the mathematical constant pi (approximately [tex]\frac{22}{7}[/tex])
• r is the radius of the cone's base
• h is the height of the cone

In this case, we have two cones with different heights:

[tex]V_{cone1} = \frac{1}{3}\times \frac{22}{7} \times(2.1 cm)^{2}\times4.1 cm[/tex]

[tex]V_{cone2} = \frac{1}{3}\times \frac{22}{7} \times(2.1 cm)^{2}\times 4.3 cm[/tex]

2. Calculate the total volume of the melted cones.

Since the cones are melted together, their volumes simply add up:

[tex]V_{total} = V_{cone1 }+ V_{cone2}[/tex]

3. Calculate the radius of the sphere.

The volume of a sphere is given by:

[tex]V_{sphere }= \frac{4}{3}\pi r^{3}\: units^{3}[/tex]

where:

• [tex]V_{sphere}[/tex] is the volume of the sphere
• r is the radius of the sphere

We want to find the radius of the sphere that has the same volume as the melted cones. Therefore, we can set Vtotal equal to Vsphere and solve for r:

[tex]V_{total }= \frac{4}{3}\pi r^{3}[/tex]

[tex]V_{cone1} = 18.942[/tex]

[tex]V_{cone2} = 19.866[/tex]

[tex]V_{total }= V_{cone1}+V_{cone2} = \frac{4}{3}\pi r^{3}[/tex]

[tex]V_{total }= 18.942+19.866 = \frac{4}{3}\pi r^{3}[/tex]

[tex]38.808 = \frac{4}{3}\pi r^{3}[/tex]

[tex]38.808 \times \frac{3}{4} \times \frac{7}{22}= r^{3}[/tex]

[tex]\rightarrow r^{3} = 9.261[/tex]

[tex]r = \sqrt[3]{9.261}\: cm[/tex]

[tex]\rightarrow r = 2.1 \:cm[/tex]

4. Calculate the diameter of the sphere.

Finally, the diameter of the sphere is simply twice its radius:

Diameter = [tex]2 \times r[/tex]

Diameter = [tex]2 \times 2.1 \:cm[/tex]

[tex]\bold{Diameter} = 4.2 \: cm \rightarrow \rightarrow Option \:(C)[/tex]

I hope this helps! Let me know if you have any questions about the steps involved.

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Hope my answer helped you!!

Please thank me and mark me as Brainliest if my answer really helped you my mate!!!

Have a happy, Pleasant and Prosperous Day ahead, My mate!!!

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